(color online) Comparisons of odd (solid-black) and even (dashed-red) dipoles for HeH²⁺ with I_p = 2.25 a.u., Z₁/Z₂ = 2 at R = 2 a.u. (the left column) and R = 4 a.u. (right), obtained by different methods. In panels (a) and (b), we show the 1D exact dipoles |〈0|x|p〉|²/ω⁴ for the continuum state |p〉 having the odd-like (corresponding to odd dipoles) or even-like (even dipoles) parity. The continuum state |p〉 has the energy Ep = ω − I_p. Here, ω = nω₀ is the assumed harmonic energy with ω₀ = 0.057 a.u. (λ = 800 nm). In panels (c) and (d), we show the dipoles |D_odd(even)(θ, ω)|²/ω⁴ with θ = 0° obtained using Eqs. (4) and (5) (the previous dipoles). In panels (e) and (f), we show those obtained using Eqs. (6) and (7) (the improved dipoles). The log₁₀ scale is used here.